AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 79(3) (2021), Pages 454–460

3 A note on coloring of 3-power of subquartic graphs

Mahsa Mozafari-Nia Moharram N. Iradmusa

Department of Mathematical Sciences Shahid Beheshti University G.C. P.O. Box 19839-63113, Tehran Iran

Abstract

1 For any k ∈ N, the k-subdivision of a graph G is a simple graph G k , which is constructed by replacing each edge of G with a path of length k. In [M.N. Iradmusa, Discrete Math. 310 (10-11) (2010), 1551–1556], the mth power of the n-subdivision of G was introduced as a fractional m power of G, denoted by G n . F. Wang and X. Liu in [Discrete Math. 3 Algorithms Appl. 10 (3), (2018), 1850041] showed that χ(G 3 ) ≤ 7 for 3 any subcubic graph G. In this note, we prove that the 3 -power of every subquartic graph admits a proper coloring with at most nine colors. We 3 conjecture that χ(G 3 ) ≤ 2∆(G) + 1 for any graph G with maximum degree ∆(G) ≥ 2.

1 Introduction

All graphs we consider in this note are simple, finite and undirected. We men- tion some of the definitions that are referred to throughout this note, and for other necessary definitions and notation we refer the reader to a standard text-book [2]. Minimum degree, maximum degree and maximum size of cliques of a graph G are denoted by δ(G), ∆(G) and ω(G), respectively. For each vertex v ∈ V (G), the neigh- bors of v are the vertices adjacent to v in G. The neighborhood NG(v) of v is the set of all neighbors of v in G. Any vertex of degree k is called a k-vertex and any path of length k is called a k-path. Also, a path P : v1, . . . , vk is a simple path if the degrees of all vertices v2, . . . , vk−1 are 2, and a cycle C is called a loop cycle if all of its vertices are 2-vertices except one that is a 3-vertex. Let G be a graph and k be a positive integer. The k-power of G, denoted by Gk, is defined on the vertex set V (G) by adding edges joining any two distinct vertices x and y with distance at most k. Also the k-subdivision of G, denoted by 1 G k , is constructed by replacing each edge xy of G with a path of length k, say Pxy. These k-paths are called superedges. We denote a vertex by (xy)l if it belongs to Pxy and has distance l from the vertex x, where l ∈ {0, 1, 2, . . . , k}. Note that

ISSN: 2202-3518 c The author(s) M. MOZAFARI-NIA ET AL. / AUSTRALAS. J. COMBIN. 79 (3) (2021), 454–460 455

1 (xy)l = (yx)k−l, x = (xy)0 = (yx)k and y = (yx)0 = (xy)k. Any vertex (xy)0 of G k is called a terminal vertex (or briefly t-vertex) and any of the remaining vertices is called an internal vertex (or briefly i-vertex). The fractional power of graphs was first introduced in [5] as follows.

m Definition 1.1 Let G be a graph and m, n ∈ N. The graph G n is defined to be the m 1 m m-power of the n-subdivision of G. In other words G n = (G n ) .

m m We denote the set of terminal vertices of G n by Vt(G n ) and the set of internal m 1 2 vertices by Vi(G n ). It is worth noting that G 1 = G and G 2 = T (G), where T (G), the total graph of G, is the the graph whose vertex set is V (G) ∪ E(G), in which two vertices are adjacent if and only if they are adjacent or incident in G [1]. As usual, a proper k-coloring of G is a mapping from V (G) to {1, . . . , k}, where any two adjacent vertices have distinct colors. The chromatic number of a graph G is the minimum integer k for which G has a proper k-coloring, and is denoted 00 2 by χ(G). By the definition of a total graph, χ (G) := χ(T (G)) = χ(G 2 ). In 1965, Behzad [1] conjectured that χ00(G) never exceeds ∆(G) + 2. By virtue of Definition 2 1.1, one can show that ω(G 2 ) = ∆(G) + 1 and the Total Coloring Conjecture can be reformulated as follows.

2 2 Conjecture 1.2 For any simple graph G, χ(G 2 ) ≤ ω(G 2 ) + 1.

There is a relation between an incidence coloring of a graph G and a vertex 3 coloring of G 3 , which is one of the motivations of this note. The concept of incidence coloring was introduced by Brualdi and Massey in 1993.

Definition 1.3 [3] Let G = (V,E) be a multigraph. An incidence of G is a pair (v, e) where v ∈ V (G), e ∈ E(G) and e is incident with v. Let I be the set of incidences of G. An incidence graph of G, denoted by I(G), has its vertex set V (I(G)) = I such that two incidences (v, e) and (w, f) are adjacent in I(G) if one of the following holds:

(1) v = w,

(2) e = f,

(3) the end-vertices of e or f are v and w.

Definition 1.4 [3] An incidence coloring σ of G is a map from I to the color set C such that adjacent incidence pairs are assigned different colors. If σ : I −→ C is an incidence coloring with |C| = k, then we say that σ is a k-incidence coloring of G. The incidence chromatic number of G, denoted χi(G), is the smallest k for which there exists a k-incidence coloring of G.

In [3], it is proved that for a graph G with maximum degree ∆, χi(G) ≤ 2∆. Also, by definition of the fractional power of a graph, the incidence graph I(G) is M. MOZAFARI-NIA ET AL. / AUSTRALAS. J. COMBIN. 79 (3) (2021), 454–460 456

3 the subgraph of G 3 induced by the set of internal vertices. So we have χi(G) = 3 3 3 3 3 χ(G 3 [Vi(G 3 )]) ≤ χ(G 3 ). In addition, the partition {Vt(G 3 ),Vi(G 3 )} of the vertices 3 of G 3 implies that

3 3 3 3 3 χ(G 3 ) ≤ χ(G 3 [Vt(G 3 )]) + χ(G 3 [Vi(G 3 )]) = χ(G) + χi(G).

3 Also, in [6] it was proved that if ∆(G) ≥ 3, then χ(G 3 ) ≤ χ(G) + χi(G) − 1. 3 In this note, we are investigating the chromatic number of G 3 . When ∆(G) = 1, 3 one can easily show that χ(G 3 ) = 4, and by applying the following theorem, which 3 was proved in [5], we can prove that χ(G 3 ) ≤ 5 for any graph G with ∆(G) = 2.

Theorem 1.5 If m, n, k ∈ N and k ≥ 3, then ( nk m nk m ≥ n 2 (i) χ(Ck ) = nk nk d nk e m < 2 , b m+1 c m n (ii) χ(Pk ) = min{m + 1, (k − 1)n + 1}.

3 In [8], Wang and Liu proved that χ(G 3 ) ≤ 7 for any subcubic graph G. Recall that a graph G is subcubic if ∆(G) ≤ 3. Recently, a simple proof of this result was given in [6] by using the following theorem about the 5-colorability of the incidences of any subcubic graph.

Theorem 1.6 [7] For any subcubic graph G, we have χi(G) ≤ 5.

The graph G is called subquartic if ∆(G) ≤ 4. Also any 4-regular graph is known as a quartic graph. The main theorem of this note is stated as follows.

3 Theorem 1.7 Let G be a subquartic graph. Then χ(G 3 ) ≤ 9.

Remark 1.8 In [3], it was conjectured that χi(G) ≤ ∆(G) + 2 for every graph G. This was disproved by Guiduli in [4] who showed that Paley graphs with sufficiently large maximum degree have incidence chromatic number at least ∆+Ω(log ∆). How- ever, this conjecture seems to hold for graphs with small maximum degree. Theorem 1.6 shows that the conjecture is true for cubic graphs. It remains an open problem whether the conjecture is true for quartic graphs. But if the conjecture holds for all quartic graphs, then easily we can prove Theorem 1.7 by use of the inequality 3 χ(G 3 ) ≤ χ(G) + χi(G) − 1. So Theorem 1.7 provides evidence that the conjecture may be true for all quartic graphs.

Considering the results for cycles, paths, subcubic and subquartic graphs, we 3 conjecture that 2∆(G) + 1 colors suffice for the proper coloring of G 3 when G is a graph with maximum degree at least two.

3 Conjecture 1.9 Let G be a graph with ∆(G) ≥ 2. Then χ(G 3 ) ≤ 2∆(G) + 1. M. MOZAFARI-NIA ET AL. / AUSTRALAS. J. COMBIN. 79 (3) (2021), 454–460 457

The clique number of the fractional power of a graph was obtained in [5] for powers 3 less than one. As mentioned in [8], one can easily show that ω(G 3 ) = ∆(G)+2 when 3 ∆(G) ≥ 2, and ω(G 3 ) = 4 when ∆(G) = 1. Therefore, if Conjecture 1.2 holds, we 00 2 3 conclude that χ (G) = χ(G 2 ) ≤ χ(G 3 ). So, the following conjecture seems strongly true.

2 3 Conjecture 1.10 For any graph G, χ(G 2 ) ≤ χ(G 3 ).

2 Proof of Theorem 1.7

For convenience, we need some notation and preliminaries. 3 Let G be a graph and consider G 3 . On each superedge Puv there are two internal vertices (uv)1 and (uv)2 which correspond to the incidences of the edge uv. We denote (uv)1 and (uv)2 by (u, v) and (v, u), respectively. We need to establish the following lemmas before the proof of Theorem 1.7.

Lemma 2.1 Let Pn : v1, v2, . . . , vn be a path of order n ≥ 5, n 6= 6, 7 and the vertices v1, v2, vn−1 and vn are respectively colored with the colors a, b, c and d from the set C = {1, 2,..., 5} that are all distinct except possibly a and d. Then we can extend 3 this partial coloring to a proper coloring of Pn with colors from C.

3 Proof To extend this partial coloring to a proper coloring of Pn , we consider 8 cases:

3 (1) a = d, n ≡ 1 (mod 4): Starting from v1, we color the vertices of Pn sequentially by colors (abec)(abec)(abec) ··· where e ∈ C \{a, b, c}.

3 (2) a = d, n ≡ 2 (mod 4): Starting from v1, we color the vertices of Pn sequentially by colors (abefc)(abec)(abec) ··· where e, f ∈ C \{a, b, c} and e 6= f.

3 (3) a = d, n ≡ 3 (mod 4): Starting from v1, we color the vertices of Pn sequentially by colors (abefc)(abefc)(abec)(abec) ··· where e, f ∈ C \{a, b, c} and e 6= f.

3 (4) a = d, n ≡ 0 (mod 4): Starting from v1, we color the vertices of Pn sequentially by colors (abce)(abce) ··· (abce)(fbca) where e, f ∈ C \{a, b, c} and e 6= f.

3 (5) a 6= d, n ≡ 1 (mod 4): Starting from v1, we color the vertices of Pn sequentially by colors (abecd)(abcd)(abcd) ··· where e ∈ C \{a, b, c, d}.

3 (6) a 6= d, n ≡ 2 (mod 4): Starting from v1, we color the vertices of Pn sequentially by colors (abecd)(abecd)(abcd)(abcd) ··· where e ∈ C \{a, b, c, d}.

3 (7) a 6= d, n ≡ 3 (mod 4): Starting from v1, we color the vertices of Pn sequentially by colors (abde)(abde) ··· (abde)(acd) where e ∈ C \{a, b, c, d}.

3 (8) a 6= d, n ≡ 0 (mod 4): Starting from v1, we color the vertices of Pn sequentially by colors (abcd)(abcd)(abcd) ··· . M. MOZAFARI-NIA ET AL. / AUSTRALAS. J. COMBIN. 79 (3) (2021), 454–460 458

Each of these colorings preserves the colors of v1, v2, vn−1 and vn and it can be easily 3 seen that the given coloring is a proper coloring of Pn with color set C. 

Lemma 2.2 Let G be a subcubic graph with δ(G) ≥ 2 and

V3 = {v ∈ V (G) | dG(v) = 3}.

3 Then χ(G 3 \ V3) ≤ 5.

Proof Each connected component of G is a cycle or a subcubic graph in which any 2-vertex belongs to a simple path or a loop cycle. Let H be a connected component 3 3 of G. If H is a cycle, then by Theorem 1.5, we can color the vertices of H 3 \V3 = H 3 with the colors C = {1, 2, 3, 4, 5}. Now let H be a component of the second type 3 in G. To find a proper coloring for H 3 \ V3, at first we identify all 2-vertices lying in simple paths and remove all 2-vertices of the loop cycles in H. Let H1 be the 3 3 3 3 resulting graph. By Theorem 1.6, χ(H1 [Vi(H1 )]) = χi(H1) ≤ 5. Suppose that 3 3 3 3 3 3 c : Vi(H1 ) −→ C is a proper coloring of H1 [Vi(H1 )]. Suppose the simple path P : v1, v2, . . . , vn in H. The vertices v2, . . . , vn−1 of P contracted to a single vertex ∗ 3 v in H1. The subgraph of H 3 \ V3 induced by

VP = {(v1, v2), (v2, v1), v2, (v2, v3), (v3, v2), v3, . . . , vn−1, (vn−1, vn), (vn, vn−1)}

3 is isomorphic to P3n−4. Now we color the first two vertices and the last two vertices of VP as follows: ∗ ∗ c(v1, v2) = c(v1, v ), c(v2, v1) = c(v , v1), ∗ ∗ c(vn, vn−1) = c(vn, v ), c(vn−1, vn) = c(v , vn).

3 Because 5 ≤ 3n−4 6= 6, 7, by Lemma 2.1 we can extend c to the other vertices of P 3 except v1 and vn and similarly, to all vertices of the other simple paths. We denote by c0 this extension of c.

Finally, we color the vertices of the loop cycles. Let L : v1, v2, . . . , vn, v1 be a loop cycle of G such that dG(v1) = 3 and NG(v1) = {v0, v2, vn}. Suppose that 3 3 0 0 3 c ((v0, v1)) = a and c ((v1, v0)) = b. Because L 3 is isomorphic to Cn , by Theo- 3 3 rem 1.5, we have χ(L 3 ) ≤ 5. Let cL be a proper coloring of L 3 such that cL(v1) = b, cL((v1, vn)) = c and cL((v1, v2)) = d such that {c, d} ⊂ C \{a, b}. Therefore, 0 cL((v1, v2)) 6= a = c ((v0, v1)) 6= cL((v1, vn)). Because cL is a proper coloring, 0 cL((v2, v1)) 6= b = c ((v1, v0)) 6= cL((vn, v1)). Now we delete the color of v1 and 3 0 add this coloring of 3 -power of L to c . By repeating this method for each loop cycle, 0 3 c gives rise to a proper coloring of G 3 \ V3. 

Proof of Theorem 1.7. Since each subquartic graph is a subgraph of a quartic graph, we only prove the theorem for quartic graphs. Let G be a quartic graph. If G 3 is a complete graph, then G = K5. In Figure 1, a proper 7-coloring of K5 3 is shown. 3 Some edges of K5 3 were removed in the figure for simplicity. Now, suppose that G M. MOZAFARI-NIA ET AL. / AUSTRALAS. J. COMBIN. 79 (3) (2021), 454–460 459

4

5 2 3 7 1 6 2 5 6 3 1 4 4 1

3 5 5 3 7 2

1 6 4 2

3 Figure 1: 7-proper coloring of K5 3 .

is not a complete graph and M1 = {e1, . . . , ek} is a maximum matching. Since M1 is a maximum matching of G, A = V (G) \ V (M1) is an independent set of G and so NG(v) ⊆ V (M1) for each v ∈ A.

Consider the bipartite subgraph H with bipartition (A, V (M1)) that contains all edges between A and V (M1). Since dH (v) = 4 > dH (u) for any vertex v ∈ A and any vertex u ∈ V (M1), by Hall’s Theorem, H has a matching, named M2, which covers all the vertices of A. Without loss of generality, suppose that M2 = {f1, . . . , fk0 }. Note that none of the edges in M1 is adjacent to two edges of M2; otherwise, M1 is not a maximum matching of G. 3 Consider the subgraph F of G induced by M1 ∪ M2 and let B1 = V (F 3 ) \ 3 (V (M1) ∩ V (M2)) and B2 = V (G 3 ) \ B1. Now consider the partition {B1,B2} of 3 3 3 V (G 3 ). We prove that χ(G 3 [B1]) ≤ 4 and χ(G 3 [B2]) ≤ 5 and then we conclude that 3 3 3 χ(G 3 ) ≤ χ(G 3 [B1]) + χ(G 3 [B2]) ≤ 9. Since G is neither a complete graph nor an odd cycle, by Brooks’ Theorem, 3 χ(G) ≤ 4. In addition, the subgraph of G 3 induced by the t-vertices is isomorphic to 3 G. Therefore, there exists a proper coloring c of the t-vertices of G 3 [B1] with colors 3 in {1, 2, 3, 4}. Now, we color the i-vertices of G 3 [B1] with colors in {1, 2, 3, 4}. For any edge e = uv ∈ M1 that is not adjacent with any edge of M2, color i-vertices (u, v) and (v, u) with two different colors of {1, 2, 3, 4}\{c(u), c(v)}. Now, suppose that e = uv ∈ M1 is adjacent with an edge f = vw ∈ M2. If c(u) 6= c(w), then color the i-vertices (u, v) and (w, v) with a same color from {1, 2, 3, 4}\{c(u), c(w)}. Also, assign colors c(u) and c(w) to i-vertices (v, w) and (v, u), respectively. If c(u) = c(w), color the i-vertices (u, v) and (w, v) with a same color from {1, 2, 3, 4}\{c(u)} and then assign two different colors in {1, 2, 3, 4}\{c(u), c((u, v))} to the i-vertices (v, w) 3 and (v, u). Therefore χ(G 3 [B1]) ≤ 4. 3 To find a proper 5-coloring of G 3 [B2], we apply Lemma 2.2. Let G1 be the spanning subgraph of G with edge set E(G1) = E(G) \ (M1 ∪ M2) and V3 be the set of 3-vertices of G1. Because dG1 (v) = 2 for any vertex v ∈ V (M1) ∩ V (M2) and dG1 (v) = 3 for any vertex v∈ / V (M1) ∩ V (M2), we have δ(G1) ≥ 2, ∆(G1) = 3 and 3 3 3 3 3 3 G1 \ V3 = G 3 [B2]. Therefore, by Lemma 2.2, we have χ(G 3 [B2]) = χ(G1 \ V3) ≤ 5, which completes the proof.  M. MOZAFARI-NIA ET AL. / AUSTRALAS. J. COMBIN. 79 (3) (2021), 454–460 460

3 Problem 1 We did not find a subquartic graph G with χ(G 3 ) = 9. Therefore, it remains an open problem as to whether the upper bound in Theorem 1.7 can be decreased to 8 colors.

Acknowledgements

We would like to thank the editors and the anonymous referees for their suggestions which help us to improve the note.

References

[1] M. Behzad, Graphs and their Chromatic Numbers, Ph.D. Thesis, Michigan State University (1965).

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[3] R. A. Brualdi and J. Q. Massey, Incidence and strong edge colorings of graphs, Discrete Math. 122 (1993), 51–58.

[4] B. Guiduli, On incidence coloring and star arboricity of graphs, Discrete Math. 163 (1997), 275–278.

[5] M. N. Iradmusa, On colorings of graph fractional powers, Discrete Math. 310 (10-11) (2010), 1551–1556.

3 [6] M. N Iradmusa, A short proof of 7-colorability of 3 -power of subcubic graphs, Iranian J. Science and Tech., Transactions A: Science 44 (1) (2020), 225–226.

[7] M. Maydanskiy, The incidence coloring conjecture for graphs of maximum degree three, Discrete Math. 292 (2005), 131–141.

[8] F. Wang and X. Liu, Coloring 3-power of 3-subdivision of subcubic graph, Discrete Math. Algorithms Appl. 10 (3) (2018), 1850041, 9 pp.

(Received 3 May 2020; revised 21 Nov 2020)